Question: Complete the square to rewrite this expression in the form $(x + a)^2 + b$. $a$ and $b$ can be positive or negative. $ x^{2}-10x+54$
Solution: If we square the sum of $x$ and another number, we get $ \begin{align} (x + a)^2 &= (x + a)(x + a) \\ &= x^2 + {2a}x + a^2\end{align}$ The number multiplying the $x$ term is $2$ times the number that was added inside the square. In the problem we're trying to solve, the number multiplying the $x$ term is $-10$ , so when we rewrite the expression, the number added inside the square will be half of $-10$ . (Which is $-5$ $ (x - 5)^2 + b$ How can we find the value of $b$ If we multiply out the square in this expression, we get $ \begin{align} (x - 5)^2 + b &= (x - 5)(x - 5) + b \\ &= x^2 - 10x + 25 + b \end{align}$ This looks just like the given expression if $ 25 + b = 54$ So $b$ must be $29$ $x^{2}-10x+54$ can be rewritten as: $ (x - 5)^2 + 29$ Now it's just a square minus another number! We have completed the square.